Search results
Let $H$ be the entropy function and $X,Y,Z$ random variables, show that $ H (X,Y) + H (Y,Z) + H (X,Z) \geq 2 H (X,Y,Z)$. I have tried to use some identities like $H (U,V) = H (U) + H (V|U)$ then. $$H (X,Y,Z) = H (X,Y)+ H (Z|X,Y)$$. $$H (X,Y,Z) = H (X,Z) + H (Y|X,Z)$$.
Mar 5, 2018 · A corollary of this theorem is: $$ H (X,Y \mid Z) = H (X \mid Z)+H (Y \mid X,Z) $$ Proof:
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. Here, information is measured in shannons, nats, or hartleys. The entropy of conditioned on is written as .
The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x = 1, the blue plane shows the points with z = 1, and the yellow plane shows the points with y = −1.
given in equation (2.3), H(Y;Z) = H(Y) + H(ZjY), except that everything is now condi-tioned on X. As we touched on before, conditioning on an event creates a new probability space where all the same concepts of probability apply. We simply added the same condi-tioning event to all three terms!
BSC. Y. crossover probability p. p = 1 − p. channel matrix w X 0 p Y 0. p p p p( x = 0) = w w 1 p p 1 p( x = 1) = 1 − w = w p p. I(X;Y) = H(Y) - H(Y|X) I(X;Y) = H(X) - H(X|Y) Mutual Information for the BSC.
Joint and Conditional Entropy. y Recall that the entropy of rv X over X, is defined byH(X) = − Q PX(x)logPX(x)x∈X Shorter notatio. of (jointly distributed) rvs X and Y with (X;Y) H(X;Y) = − Q p(x;y)logp(x;y) x;y. This is simply the entropy of the rv Z = (X;Y).
